Who among the following is regarded as the first scientific geographer to estimate the length of the equator on sound mathematical principles?

Introduction / Context:
History of geography includes many early scholars who tried to understand the size and shape of the earth. Among them, one name is especially famous for making the first reasonably accurate calculation of the earth circumference and therefore the length of the equator, using scientific and mathematical reasoning instead of guesswork. This question checks whether you can identify that pioneering scientific geographer.

Given Data / Assumptions:

• Four ancient Greek scholars are listed: Herodotus, Hecataeus, Anaximander and Eratosthenes.
• We are asked specifically about the first person to estimate the length of the equator on sound principles.
• We assume the widely accepted historical account of early geographical thought.

Concept / Approach:
Eratosthenes of Cyrene, a Greek scholar working in Alexandria, is celebrated as the father of scientific geography. He used observations of the sun shadow at two different locations, Syene and Alexandria, on the day of the summer solstice. By measuring the angle difference in the shadows and knowing the distance between the two cities, he applied simple geometry to estimate the full circumference of the earth. From that, the length of the equator could be inferred. This method was far ahead of his time and produced a value surprisingly close to modern measurements, especially considering the limited tools available.

Step-by-Step Solution:
Step 1: Recall that Herodotus and Hecataeus are mainly known as early historians and descriptive geographers, not for mathematical measurements of the earth size.Step 2: Remember that Anaximander contributed to early cosmological ideas and map making but did not carry out accurate numerical calculations of the earth circumference.Step 3: Recognise that Eratosthenes used observations of shadows cast by the sun at two distant points along the same meridian.Step 4: He measured the angular difference and used proportional reasoning to estimate the total circumference of the earth.Step 5: Since the equator is a great circle of the earth, this calculation directly relates to its length, which is why Eratosthenes is credited as the first scientific geographer to do this.

Verification / Alternative check:
Standard geography and history of science books repeatedly mention Eratosthenes in connection with the first systematic measurement of earth circumference. They explain his use of wells, shadows and geometry, and often provide diagrams of his method. The other three scholars are discussed for different contributions, such as early world maps or descriptive accounts of lands and peoples, but not for this specific achievement. This repeated emphasis confirms that Eratosthenes is the correct choice.

Why Other Options Are Wrong:
Herodotus: Mainly known as the father of history, providing descriptive geographical information but not scientific measurements of earth size.Hecataeus: An early geographer and cartographer, but he did not make a rigorous calculation of the equator length.Anaximander: Proposed early cosmological models and produced maps, yet he did not perform the precise mathematical estimation achieved by Eratosthenes.

Common Pitfalls:
Candidates sometimes remember only that several Greek scholars contributed to geography and choose randomly. Others may pick Herodotus because they recognise his name more easily from history. To avoid this, it is useful to link Eratosthenes specifically with three key ideas: scientific geography, measurement of earth circumference, and work at the Library of Alexandria. These associations will quickly guide you to the correct answer during exams.

Final Answer:
The first scientific geographer who estimated the length of the equator on sound mathematical principles was Eratosthenes.